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Rectifiable metric spaces: Local structure and regularity of the Hausdorff measure. (English) Zbl 0806.28004
The author considers $$n$$-rectifiable metric spaces $$(X,\rho)$$. He shows analogously to the Euclidean setting: For almost all $$x\in X$$ with respect to the $$n$$-dimensional Hausdorff measure $${\mathcal H}^ n_ \rho$$ (understood as in Federer’s book) there exist a norm $$\|\cdot\|_ x$$ on $$\mathbb{R}^ n$$, a map $$\phi_ x: X\to \mathbb{R}^ n$$, and a closed set $$A_ x\subset X$$ such that $$\phi_ x(x)= 0$$, $\lim_{r\to 0} (\alpha(n) r^ n)^{-1} {\mathcal H}^ n_ \rho(B_ \rho(x,r)\cap A_ x)= 1,$ and $\limsup_{r\to 0} \left\{\left|1- {\|\phi_ x(y)- \phi_ x(z)\|_ x\over \rho(y,z)}\right|;\;y,z\in A_ x\cap B_ \rho(x,r),\;y\neq z\right\}= 0.$ He gives also an extension of ideas due to Kolmogoroff concerning maximal and minimal strongly metrically invariant measures. He has found an integral representation of these measures on the space $$C([0,1])$$. As a consequence, the equivalence class of Banach-Mazur compactum containing $$\|\cdot\|_ x$$ consists of all norms on $$\mathbb{R}^ n$$ induced by an inner product provided that the maximal and minimal Kolmogoroff measures coincide.

##### MSC:
 28A78 Hausdorff and packing measures 26B10 Implicit function theorems, Jacobians, transformations with several variables
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##### References:
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